text: from core import C
  from sympify import converter, sympify, _sympify, SympifyError
  from basic import Basic
  from singleton import S, Singleton
  from expr import Expr, AtomicExpr
  from decorators import _sympifyit, deprecated
  from cache import cacheit, clear_cache
  from sympy.core.compatibility import as_int
  import sympy.mpmath as mpmath
  import sympy.mpmath.libmp as mlib
  from sympy.mpmath.libmp import mpf_pow, mpf_pi, mpf_e, phi_fixed
  from sympy.mpmath.ctx_mp import mpnumeric
  from sympy.mpmath.libmp.libmpf import fnan, _normalize as mpf_normalize
  
  import decimal
  import math
  import re as regex
  from collections import defaultdict
  
  rnd = mlib.round_nearest
  
  _LOG2 = math.log(2)
  
  def mpf_norm(mpf, prec):
      """Return the mpf tuple normalized appropriately for the indicated
      precision.
  
      This also contains a portion of code to not return zero if
      the mantissa is 0 since it is zero for mpf's +inf, -inf and
      nan, too.
      """
      sign, man, expt, bc = mpf
      if not man:
          # hack for mpf_normalize which does not do this;
          # it assumes that if man is zero the result is 0
          if not bc:
              return mlib.fzero
          else:
              # don't change anything; this should already
              # be a well formed mpf tuple
              return mpf
      rv = mpf_normalize(sign, man, expt, bc, prec, rnd)
      return rv
  
  # TODO: we should use the warnings module
  _errdict = {"divide": False}
  def seterr(divide=False):
      """
      Should sympy raise an exception on 0/0 or return a nan?
  
      divide == True .... raise an exception
      divide == False ... return nan
      """
      if _errdict["divide"] != divide:
          clear_cache()
          _errdict["divide"] = divide
  
  def _decimal_to_Rational_prec(dec):
      """Convert an ordinary decimal instance to a Rational."""
      # _is_special is needed for Python 2.5 support; is_finite for Python 3.3
      # support
      nonfinite = getattr(dec, '_is_special', None)
      if nonfinite is None:
          nonfinite = not dec.is_finite()
      if nonfinite:
          raise TypeError("dec must be finite, got %s." % dec)
      s, d, e = dec.as_tuple()
      prec = len(d)
      if int(dec) == dec:
          rv = Rational(int(dec))
      else:
          s = (-1)**s
          d = sum([di*10**i for i, di in enumerate(reversed(d))])
          rv = Rational(s*d, 10**-e)
      return rv, prec
  
  def _literal_float(f):
      """Return True if n can be interpreted as a floating point number."""
      pat = r"[-+]?((\d*\.\d+)|(\d+\.?))(eE[-+]?\d+)?"
      return bool(regex.match(pat, f))
  
  # (a,b) -> gcd(a,b)
  _gcdcache = {}
  
  # TODO caching with decorator, but not to degrade performance
  def igcd(a, b):
      """Computes positive, integer greatest common divisor of two numbers.
  
         The algorithm is based on the well known Euclid's algorithm. To
         improve speed, igcd() has its own caching mechanism implemented.
      """
      try:
          return _gcdcache[(a,b)]
      except KeyError:
          a, b = as_int(a), as_int(b)
  
          if a and b:
              if b < 0:
                  b = -b
  
              while b:
                  a, b = b, a % b
          else:
              a = abs(a or b)
  
          _gcdcache[(a,b)] = a
          return a
  
  def ilcm(a, b):
      """Computes integer least common multiple of two numbers. """
      if a == 0 and b == 0:
          return 0
      else:
          return a*b // igcd(a, b)
  
  def igcdex(a, b):
      """Returns x, y, g such that g = x*a + y*b = gcd(a, b).
  
         >>> from sympy.core.numbers import igcdex
         >>> igcdex(2, 3)
         (-1, 1, 1)
         >>> igcdex(10, 12)
         (-1, 1, 2)
  
         >>> x, y, g = igcdex(100, 2004)
         >>> x, y, g
         (-20, 1, 4)
         >>> x*100 + y*2004
         4
  
      """
      if (not a) and (not b):
          return (0, 1, 0)
  
      if not a:
          return (0, b//abs(b), abs(b))
      if not b:
          return (a//abs(a), 0, abs(a))
  
      if a < 0:
          a, x_sign = -a, -1
      else:
          x_sign = 1
  
      if b < 0:
          b, y_sign = -b, -1
      else:
          y_sign = 1
  
      x, y, r, s = 1, 0, 0, 1
  
      while b:
          (c, q) = (a % b, a // b)
          (a, b, r, s, x, y) = (b, c, x-q*r, y-q*s, r, s)
  
      return (x*x_sign, y*y_sign, a)
  
  class Number(AtomicExpr):
      """
      Represents any kind of number in sympy.
  
      Floating point numbers are represented by the Float class.
      Integer numbers (of any size), together with rational numbers (again,
      there is no limit on their size) are represented by the Rational class.
  
      If you want to represent, for example, ``1+sqrt(2)``, then you need to do::
  
        Rational(1) + sqrt(Rational(2))
      """
      is_commutative = True
      is_bounded = True
      is_finite = True
      is_number = True
  
      __slots__ = []
  
      # Used to make max(x._prec, y._prec) return x._prec when only x is a float
      _prec = -1
  
      is_Number = True
  
      def __new__(cls, *obj):
          if len(obj) == 1:
              obj=obj[0]
  
          if isinstance(obj, Number):
              return obj
          if isinstance(obj, (int, long)):
              return Integer(obj)
          if isinstance(obj, tuple) and len(obj) == 2:
              return Rational(*obj)
          if isinstance(obj, (float, mpmath.mpf, decimal.Decimal)):
              return Float(obj)
          if isinstance(obj, basestring):
              val = sympify(obj)
              if isinstance(val, Number):
                  return val
              else:
                  raise ValueError('String "%s" does not denote a Number'%obj)
              if isinstance(obj, Number):
                  return obj
          msg = "expected str|int|long|float|Decimal|Number object but got %r"
          raise TypeError(msg % type(obj).__name__)
  
      def __divmod__(self, other):
          from containers import Tuple
          from sympy.functions.elementary.complexes import sign
  
          try:
              other = Number(other)
          except TypeError:
              msg = "unsupported operand type(s) for divmod(): '%s' and '%s'"
              raise TypeError(msg % (type(self).__name__, type(other).__name__))
          if not other:
              raise ZeroDivisionError('modulo by zero')
          if self.is_Integer and other.is_Integer:
              return Tuple(*divmod(self.p, other.p))
          else:
              rat = self/other
          w = sign(rat)*int(abs(rat)) # = rat.floor()
          r = self - other*w
          #w*other + r == self
          return Tuple(w, r)
  
      def __rdivmod__(self, other):
          try:
              other = Number(other)
          except TypeError:
              msg = "unsupported operand type(s) for divmod(): '%s' and '%s'"
              raise TypeError(msg % (type(other).__name__, type(self).__name__))
          return divmod(other, self)
  
      def __round__(self, *args):
          return round(float(self), *args)
  
      def _as_mpf_val(self, prec):
          """Evaluation of mpf tuple accurate to at least prec bits."""
          raise NotImplementedError('%s needs ._as_mpf_val() method' % \
              (self.__class__.__name__))
  
      def _eval_evalf(self, prec):
          return Float._new(self._as_mpf_val(prec), prec)
  
      def _as_mpf_op(self, prec):
          prec = max(prec, self._prec)
          return self._as_mpf_val(prec), prec
  
      def __float__(self):
          return mlib.to_float(self._as_mpf_val(53))
  
      def _eval_conjugate(self):
          return self
  
      def _eval_order(self, *symbols):
          # Order(5, x, y) -> Order(1,x,y)
          return C.Order(S.One, *symbols)
  
      def _eval_subs(self, old, new):
          if old == -self:
              return -new
          return self # there is no other possibility
  
      @classmethod
      def class_key(cls):
          return 1, 0, 'Number'
  
      @cacheit
      def sort_key(self, order=None):
          return self.class_key(), (0, ()), (), self
  
      @_sympifyit('other', NotImplemented)
      def __add__(self, other):
          if isinstance(other, Number):
              if other is S.NaN:
                  return S.NaN
              elif other is S.Infinity:
                  return S.Infinity
              elif other is S.NegativeInfinity:
                  return S.NegativeInfinity
          return AtomicExpr.__add__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __sub__(self, other):
          if isinstance(other, Number):
              if other is S.NaN:
                  return S.NaN
              elif other is S.Infinity:
                  return S.NegativeInfinity
              elif other is S.NegativeInfinity:
                  return S.Infinity
          return AtomicExpr.__sub__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __mul__(self, other):
          if isinstance(other, Number):
              if other is S.NaN:
                  return S.NaN
              elif other is S.Infinity:
                  if self == 0:
                      return S.NaN
                  elif self > 0:
                      return S.Infinity
                  else:
                      return S.NegativeInfinity
              elif other is S.NegativeInfinity:
                  if self == 0:
                      return S.NaN
                  elif self > 0:
                      return S.NegativeInfinity
                  else:
                      return S.Infinity
          return AtomicExpr.__mul__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __div__(self, other):
          if isinstance(other, Number):
              if other is S.NaN:
                  return S.NaN
              elif other is S.Infinity or other is S.NegativeInfinity:
                  return S.Zero
          return AtomicExpr.__div__(self, other)
  
      __truediv__ = __div__
  
  
      def __eq__(self, other):
          raise NotImplementedError('%s needs .__eq__() method' %
              (self.__class__.__name__))
  
      def __ne__(self, other):
          raise NotImplementedError('%s needs .__ne__() method' %
              (self.__class__.__name__))
  
      def __lt__(self, other):
          raise NotImplementedError('%s needs .__lt__() method' %
              (self.__class__.__name__))
  
      def __le__(self, other):
          raise NotImplementedError('%s needs .__le__() method' %
              (self.__class__.__name__))
  
      def __gt__(self, other):
          return _sympify(other).__lt__(self)
  
      def __ge__(self, other):
          return _sympify(other).__le__(self)
  
      def __hash__(self):
          return super(Number, self).__hash__()
  
      def is_constant(self, *wrt, **flags):
          return True
  
      @property
      def is_number(self):
          return True
  
      def as_coeff_mul(self, *deps):
          # a -> c*t
          if self.is_Rational:
              return self, tuple()
          elif self.is_negative:
              return S.NegativeOne, (-self,)
          return S.One, (self,)
  
      def as_coeff_add(self, *deps):
          # a -> c + t
          if self.is_Rational:
              return self, tuple()
          return S.Zero, (self,)
  
      def gcd(self, other):
          """Compute greatest common divisor of input arguments. """
          return S.One
  
      def lcm(self, other):
          """Compute least common multiple of input arguments. """
          other = _sympify(other)
          return self*other
  
      def cofactors(self, other):
          """Compute GCD and cofactors of input arguments. """
          other = _sympify(other)
          return S.One, self, other
  
      def as_coeff_Mul(self, rational=False):
          """Efficiently extract the coefficient of a product. """
          if rational and not self.is_Rational:
              return S.One, self
          return self, S.One
  
      def as_coeff_Add(self):
          """Efficiently extract the coefficient of a summation. """
          return self, S.Zero
  
  class Float(Number):
      """
      Represents a floating point number. It is capable of representing
      arbitrary-precision floating-point numbers.
  
      Examples
      ========
  
      >>> from sympy import Float
      >>> Float(3.5) # convert from Python float or int
      3.50000000000000
      >>> Float(3) # reverts to Integer
      3
      >>> Float(3, '') # forced to Float
      3.
  
      Floats can be created from a string representations of Python floats
      to force ints to Float or to enter high-precision (> 15 significant
      digits) values:
  
      >>> Float('.0010')
      0.00100000000000000
      >>> Float('1e-3')
      0.00100000000000000
      >>> Float('1e-3', 3)
      0.00100
  
      Float can automatically count significant figures if a null string
      is sent for the precision; space are also allowed in the string. (Auto-
      counting is only allowed for strings, ints and longs).
  
      >>> Float('123 456 789 . 123 456', '')
      123456789.123456
      >>> Float('12e-3', '')
      0.012
  
      Notes
      =====
  
      Floats are inexact by their nature unless their value is a binary-exact
      value.
  
      >>> approx, exact = Float(.1, 1), Float(.125, 1)
  
      For calculation purposes, evalf needs to be able to change the precision
      but this will not increase the accuracy of the inexact value. The
      following is the most accurate 5-digit approximation of a value of 0.1
      that had only 1 digit of precision:
  
      >>> approx.evalf(5)
      0.099609
  
      By contrast, 0.125 is exact in binary (as it is in base 10) and so it
      can be passed to Float or evalf to obtain an arbitrary precision with
      matching accuracy:
  
      >>> Float(exact, 5)
      0.12500
      >>> exact.evalf(20)
      0.12500000000000000000
  
      Trying to make a high-precision Float from a float is not disallowed,
      but one must keep in mind that the *underlying float* (not the apparent
      decimal value) is being obtained with high precision. For example, 0.3
      does not have a finite binary representation. The closest rational is
      the fraction 5404319552844595/2**54. So if you try to obtain a Float of
      0.3 to 20 digits of precision you will not see the same thing as 0.3
      followed by 19 zeros:
  
      >>> Float(0.3, 20)
      0.29999999999999998890
  
      If you want a 20-digit value of the decimal 0.3 (not the floating point
      approximation of 0.3) you should send the 0.3 as a string. The underlying
      representation is still binary but a higher precision than Python's float
      is used:
  
      >>> Float('0.3', 20)
      0.30000000000000000000
  
      Although you can increase the precision of an existing Float using Float
      it will not increase the accuracy -- the underlying value is not changed:
  
      >>> def show(f): # binary rep of Float
      ...     from sympy import Mul, Pow
      ...     s, m, e, b = f._mpf_
      ...     v = Mul(m, Pow(2, e, evaluate=False), evaluate=False)
      ...     print '%s at prec=%s' % (v, f._prec)
      ...
      >>> t = Float('0.3', 3)
      >>> show(t)
      4915/2**14 at prec=13
      >>> show(Float(t, 20)) # higher prec, not higher accuracy
      4915/2**14 at prec=70
      >>> show(Float(t, 2)) # lower prec
      307/2**10 at prec=10
  
      The same thing happens when evalf is used on a Float:
  
      >>> show(t.evalf(20))
      4915/2**14 at prec=70
      >>> show(t.evalf(2))
      307/2**10 at prec=10
  
      Finally, Floats can be instantiated with an mpf tuple (n, c, p) to
      produce the number (-1)**n*c*2**p:
  
      >>> n, c, p = 1, 5, 0
      >>> (-1)**n*c*2**p
      -5
      >>> Float((1, 5, 0))
      -5.00000000000000
  
      An actual mpf tuple also contains the number of bits in c as the last
      element of the tuple, but this is not needed for instantiation:
  
          >>> _._mpf_
          (1, 5, 0, 3)
  
      """
      __slots__ = ['_mpf_', '_prec']
  
      is_real = True
      is_irrational = False
      is_integer = False
  
      is_Float = True
  
      def __new__(cls, num, prec=15):
          if isinstance(num, basestring):
              num = num.replace(' ', '')
              if num.startswith('.') and len(num) > 1:
                  num = '0' + num
              elif num.startswith('-.') and len(num) > 2:
                  num = '-0.' + num[2:]
          elif not num:
              return C.Zero()
          if prec == '':
              if isinstance(num, (int, long, Integer)):
                  # an int is unambiguous, but if someone enters
                  # .99999999999999999, Python automatically converts
                  # this to 1.0 and although 1.0 == 1, this is not
                  # really what the user typed, so we avoid guessing --
                  # even if num == int(num) -- because we don't know how
                  # it became that exact float.
                  num = str(num)
              elif not isinstance(num, basestring):
                  raise ValueError('The null string can only be used when '
                  'the number to Float is passed as a string.')
              ok = None
              if _literal_float(num):
                  try:
                      Num = decimal.Decimal(num)
                  except decimal.InvalidOperation:
                      pass
                  else:
                      num, dps = _decimal_to_Rational_prec(Num)
                      ok = True
                      if num.is_Integer:
                          dps = len(str(num))
              if ok is None:
                  raise ValueError('string-float not recognized: %s' % num)
          else:
              dps = prec
  
          if prec != '' and isinstance(num, (int, long, Integer)):
              # if this is changed here it has to be changed in _new, too
              return Integer(num)
  
          prec = mlib.libmpf.dps_to_prec(dps)
          if isinstance(num, float):
              _mpf_ = mlib.from_float(num, prec, rnd)
          elif isinstance(num, (str, decimal.Decimal, Integer)):
              _mpf_ = mlib.from_str(str(num), prec, rnd)
          elif isinstance(num, Rational):
              _mpf_ = mlib.from_rational(num.p, num.q, prec, rnd)
          elif isinstance(num, tuple) and len(num) in (3, 4):
              if type(num[1]) is str:
                  # it's a hexadecimal (coming from a pickled object)
                  # assume that it is in standard form
                  num = list(num)
                  num[1] = long(num[1], 16)
                  _mpf_ = tuple(num)
              else:
                  if not num[1] and len(num) == 4:
                      # handle normalization hack
                      return Float._new(num, prec)
                  else:
                      _mpf_ = mpmath.mpf(
                      S.NegativeOne**num[0]*num[1]*2**num[2])._mpf_
          elif isinstance(num, Float):
              _mpf_ = num._mpf_
              if prec < num._prec:
                  _mpf_ = mpf_norm(_mpf_, prec)
          else:
              _mpf_ = mpmath.mpf(num)._mpf_
  
          if not num:
              return C.Zero()
  
          obj = Expr.__new__(cls)
          obj._mpf_ = _mpf_
          obj._prec = prec
          return obj
  
      @classmethod
      def _new(cls, _mpf_, _prec):
          if _mpf_ == mlib.fzero:
              return S.Zero
  
          # the new Float should be normalized unless it is
          # an integer because Float doesn't return Floats
          # for Integers. If Integers can become Floats then
          # all the following (up to the first 'obj =' line
          # can be replaced with ok = mpf_norm(_mpf_, _prec)
  
          sign, man, expt, bc = _mpf_
          if not man:
              # hack for mpf_normalize which does not do this
              if not bc:
                  ok = mlib.fzero
              else:
                  ok = (sign % 2, long(man), expt, bc)
          elif expt < 0:
              # this is the non-hack normalization
              ok = mpf_normalize(sign, man, expt, bc, _prec, rnd)
          else:
              ok = _mpf_
  
          obj = Expr.__new__(cls)
          obj._mpf_ = ok
          obj._prec = _prec
          return obj
  
      # mpz can't be pickled
      def __getnewargs__(self):
          return (mlib.to_pickable(self._mpf_),)
  
      def __getstate__(self):
          return {'_prec': self._prec}
  
      def _hashable_content(self):
          return (self._mpf_, self._prec)
  
      def floor(self):
          return C.Integer(int(mlib.to_int(
              mlib.mpf_floor(self._mpf_, self._prec))))
  
      def ceiling(self):
          return C.Integer(int(mlib.to_int(
              mlib.mpf_ceil(self._mpf_, self._prec))))
  
      @property
      def num(self):
          return mpmath.mpf(self._mpf_)
  
      def _as_mpf_val(self, prec):
          rv = mpf_norm(self._mpf_, prec)
          # uncomment to see failures
          #if rv != was._mpf_ and self._prec == prec:
          #    print was._mpf_, rv
          return rv
  
      def _as_mpf_op(self, prec):
          return self._mpf_, max(prec, self._prec)
  
      def _eval_is_positive(self):
          return self.num > 0
  
      def _eval_is_negative(self):
          return self.num < 0
  
      def __neg__(self):
          return Float._new(mlib.mpf_neg(self._mpf_), self._prec)
  
      @_sympifyit('other', NotImplemented)
      def __add__(self, other):
          if isinstance(other, Number):
              rhs, prec = other._as_mpf_op(self._prec)
              return Float._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec)
          return Number.__add__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __sub__(self, other):
          if isinstance(other, Number):
              rhs, prec = other._as_mpf_op(self._prec)
              return Float._new(mlib.mpf_sub(self._mpf_, rhs, prec, rnd), prec)
          return Number.__sub__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __mul__(self, other):
          if isinstance(other, Number):
              rhs, prec = other._as_mpf_op(self._prec)
              return Float._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec)
          return Number.__mul__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __div__(self, other):
          if isinstance(other, Number):
              rhs, prec = other._as_mpf_op(self._prec)
              return Float._new(mlib.mpf_div(self._mpf_, rhs, prec, rnd), prec)
          return Number.__div__(self, other)
  
      __truediv__ = __div__
  
      @_sympifyit('other', NotImplemented)
      def __mod__(self, other):
          if isinstance(other, Number):
              rhs, prec = other._as_mpf_op(self._prec)
              return Float._new(mlib.mpf_mod(self._mpf_, rhs, prec, rnd), prec)
          return Number.__mod__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __rmod__(self, other):
          if isinstance(other, Number):
              rhs, prec = other._as_mpf_op(self._prec)
              return Float._new(mlib.mpf_mod(rhs, self._mpf_, prec, rnd), prec)
          return Number.__rmod__(self, other)
  
      def _eval_power(self, expt):
          """
          expt is symbolic object but not equal to 0, 1
  
          (-p)**r -> exp(r*log(-p)) -> exp(r*(log(p) + I*Pi)) ->
                    -> p**r*(sin(Pi*r) + cos(Pi*r)*I)
          """
          if isinstance(expt, Number):
              if isinstance(expt, Integer):
                  prec = self._prec
                  return Float._new(
                      mlib.mpf_pow_int(self._mpf_, expt.p, prec, rnd), prec)
              expt, prec = expt._as_mpf_op(self._prec)
              self = self._mpf_
              try:
                  y = mpf_pow(self, expt, prec, rnd)
                  return Float._new(y, prec)
              except mlib.ComplexResult:
                  re, im = mlib.mpc_pow(
                      (self, mlib.fzero), (expt, mlib.fzero), prec, rnd)
                  return Float._new(re, prec) + \
                         Float._new(im, prec)*S.ImaginaryUnit
  
      def __abs__(self):
          return Float._new(mlib.mpf_abs(self._mpf_), self._prec)
  
      def __int__(self):
          return int(mlib.to_int(self._mpf_)) # uses round_fast = round_down
  
      def __eq__(self, other):
          if isinstance(other, float):
              # coerce to Float at same precision
              o = Float(other)
              try:
                  ompf = o._as_mpf_val(self._prec)
              except ValueError:
                  return False
              return bool(mlib.mpf_eq(self._mpf_, ompf))
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy != other  -->  not ==
          if isinstance(other, NumberSymbol):
              if other.is_irrational:
                  return False
              return other.__eq__(self)
          if isinstance(other, Float):
              # hack for the nan == nan case which, to mpf_eq is not equal
              # but to SymPy should be equal
              if other._mpf_ == self._mpf_ == fnan:
                  return True
              return bool(mlib.mpf_eq(self._mpf_, other._mpf_))
          if isinstance(other, Number):
              # numbers should compare at the same precision;
              # all _as_mpf_val routines should be sure to abide
              # by the request to change the prec if necessary; if
              # they don't, the equality test will fail since it compares
              # the mpf tuples
              ompf = other._as_mpf_val(self._prec)
              return bool(mlib.mpf_eq(self._mpf_, ompf))
          return False    # Float != non-Number
  
      def __ne__(self, other):
          return not self.__eq__(other)
  
      def __gt__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other
          if isinstance(other, NumberSymbol):
              return other.__le__(self)
          if other.is_comparable:
              other = other.evalf()
          if isinstance(other, Number):
              return bool(mlib.mpf_gt(self._mpf_, other._as_mpf_val(self._prec)))
          return Expr.__gt__(self, other)
  
      def __ge__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  -->  ! <=
          if isinstance(other, NumberSymbol):
              return other.__lt__(self)
          if other.is_comparable:
              other = other.evalf()
          if isinstance(other, Number):
              return bool(mlib.mpf_ge(self._mpf_, other._as_mpf_val(self._prec)))
          return Expr.__ge__(self, other)
  
      def __lt__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other
          if isinstance(other, NumberSymbol):
              return other.__ge__(self)
          if other.is_real and other.is_number:
              other = other.evalf()
          if isinstance(other, Number):
              return bool(mlib.mpf_lt(self._mpf_, other._as_mpf_val(self._prec)))
          return Expr.__lt__(self, other)
  
      def __le__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  -->  ! <=
          if isinstance(other, NumberSymbol):
              return other.__gt__(self)
          if other.is_real and other.is_number:
              other = other.evalf()
          if isinstance(other, Number):
              return bool(mlib.mpf_le(self._mpf_, other._as_mpf_val(self._prec)))
          return Expr.__le__(self, other)
  
      def __hash__(self):
          return super(Float, self).__hash__()
  
      def epsilon_eq(self, other, epsilon="10e-16"):
          return abs(self - other) < Float(epsilon)
  
      def _sage_(self):
          import sage.all as sage
          return sage.RealNumber(str(self))
  
  # Add sympify converters
  converter[float] = converter[decimal.Decimal] = Float
  
  # this is here to work nicely in Sage
  RealNumber = Float
  
  @deprecated(useinstead="Float", issue=1721, deprecated_since_version="0.7.0")
  def Real(*args, **kwargs):  # pragma: no cover
      """Deprecated alias for the Float constructor."""
      return Float(*args, **kwargs)
  
  class Rational(Number):
      """Represents integers and rational numbers (p/q) of any size.
  
      Examples
      ========
  
      >>> from sympy import Rational
      >>> from sympy.abc import x, y
      >>> Rational(3)
      3
      >>> Rational(1,2)
      1/2
      >>> Rational(1.5)
      1
  
      Rational can also accept strings that are valid literals for reals:
  
      >>> Rational("1.23")
      123/100
      >>> Rational('1e-2')
      1/100
      >>> Rational(".1")
      1/10
  
      Parsing needs for any other type of string for which a Rational is desired
      can be handled with the rational=True option in sympify() which produces
      rationals from strings like '.[3]' (=1/3) and '3/10' (=3/10).
  
      **Low-level**
  
      Access numerator and denominator as .p and .q:
  
      >>> r = Rational(3,4)
      >>> r
      3/4
      >>> r.p
      3
      >>> r.q
      4
  
      Note that p and q return integers (not sympy Integers) so some care
      is needed when using them in expressions:
  
      >>> r.p//r.q
      0
  
      """
      is_real = True
      is_integer = False
      is_rational = True
  
      __slots__ = ['p', 'q']
  
      is_Rational = True
  
      @cacheit
      def __new__(cls, p, q=None):
          if q is None:
              if isinstance(p, Rational):
                  return p
              if isinstance(p, basestring):
                  p = p.replace(' ', '')
                  try:
                      # we might have a Float
                      neg_pow, digits, expt = decimal.Decimal(p).as_tuple()
                      p = [1, -1][neg_pow]*int("".join(str(x) for x in digits))
                      if expt > 0:
                          # TODO: this branch needs a test
                          return Rational(p*Pow(10, expt), 1)
                      return Rational(p, Pow(10, -expt))
                  except decimal.InvalidOperation:
                      f = regex.match('^([-+]?[0-9]+)/([0-9]+)$', p)
                      if f:
                          n, d = f.groups()
                          return Rational(int(n), int(d))
                      raise ValueError('invalid literal: %s' % p)
              elif not isinstance(p, Basic):
                  return Rational(S(p))
              q = S.One
          if isinstance(q, Rational):
              p *= q.q
              q = q.p
          if isinstance(p, Rational):
              q *= p.q
              p = p.p
          p = int(p)
          q = int(q)
          if q == 0:
              if p == 0:
                  if _errdict["divide"]:
                      raise ValueError("Indeterminate 0/0")
                  else:
                      return S.NaN
              if p < 0:
                  return S.NegativeInfinity
              return S.Infinity
          if q < 0:
              q = -q
              p = -p
          n = igcd(abs(p), q)
          if n > 1:
              p //= n
              q //= n
          if q == 1:
              return Integer(p)
          if p == 1 and q == 2:
              return S.Half
          obj = Expr.__new__(cls)
          obj.p = p
          obj.q = q
          #obj._args = (p, q)
          return obj
  
      def limit_denominator(self, max_denominator=1000000):
          """Closest Rational to self with denominator at most max_denominator.
  
          >>> from sympy import Rational
          >>> Rational('3.141592653589793').limit_denominator(10)
          22/7
          >>> Rational('3.141592653589793').limit_denominator(100)
          311/99
  
          """
          # Algorithm notes: For any real number x, define a *best upper
          # approximation* to x to be a rational number p/q such that:
          #
          #   (1) p/q >= x, and
          #   (2) if p/q > r/s >= x then s > q, for any rational r/s.
          #
          # Define *best lower approximation* similarly.  Then it can be
          # proved that a rational number is a best upper or lower
          # approximation to x if, and only if, it is a convergent or
          # semiconvergent of the (unique shortest) continued fraction
          # associated to x.
          #
          # To find a best rational approximation with denominator <= M,
          # we find the best upper and lower approximations with
          # denominator <= M and take whichever of these is closer to x.
          # In the event of a tie, the bound with smaller denominator is
          # chosen.  If both denominators are equal (which can happen
          # only when max_denominator == 1 and self is midway between
          # two integers) the lower bound---i.e., the floor of self, is
          # taken.
  
          if max_denominator < 1:
              raise ValueError("max_denominator should be at least 1")
          if self.q <= max_denominator:
              return self
  
          p0, q0, p1, q1 = 0, 1, 1, 0
          n, d = self.p, self.q
          while True:
              a = n//d
              q2 = q0+a*q1
              if q2 > max_denominator:
                  break
              p0, q0, p1, q1 = p1, q1, p0+a*p1, q2
              n, d = d, n-a*d
  
          k = (max_denominator-q0)//q1
          bound1 = Rational(p0+k*p1, q0+k*q1)
          bound2 = Rational(p1, q1)
          if abs(bound2 - self) <= abs(bound1-self):
              return bound2
          else:
              return bound1
  
      def __getnewargs__(self):
          return (self.p, self.q)
  
      def _hashable_content(self):
          return (self.p, self.q)
  
      def _eval_is_positive(self):
          return self.p > 0
  
      def _eval_is_zero(self):
          return self.p == 0
  
      def __neg__(self):
          return Rational(-self.p, self.q)
  
      @_sympifyit('other', NotImplemented)
      def __add__(self, other):
          if isinstance(other, Rational):
              return Rational(self.p*other.q + self.q*other.p, self.q*other.q)
          elif isinstance(other, Float):
              return other + self
          else:
              return Number.__add__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __sub__(self, other):
          if isinstance(other, Rational):
              return Rational(self.p*other.q - self.q*other.p, self.q*other.q)
          elif isinstance(other, Float):
              return -other + self
          else:
              return Number.__sub__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __mul__(self, other):
          if isinstance(other, Rational):
              return Rational(self.p*other.p, self.q*other.q)
          elif isinstance(other, Float):
              return other*self
          else:
              return Number.__mul__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __div__(self, other):
          if isinstance(other, Rational):
              return Rational(self.p*other.q, self.q*other.p)
          elif isinstance(other, Float):
              return self*(1/other)
          else:
              return Number.__div__(self, other)
  
      __truediv__ = __div__
  
      @_sympifyit('other', NotImplemented)
      def __mod__(self, other):
          if isinstance(other, Rational):
              n = (self.p*other.q) // (other.p*self.q)
              return Rational(self.p*other.q - n*other.p*self.q, self.q*other.q)
          if isinstance(other, Float):
              return self.evalf() % other
          return Number.__mod__(self, other)
  
      @_sympifyit('other', NotImplemented)
      def __rmod__(self, other):
          if isinstance(other, Rational):
              return Rational.__mod__(other, self)
          if isinstance(other, Float):
              return other % self.evalf()
          return Number.__rmod__(self, other)
  
      def _eval_power(self, expt):
          if isinstance(expt, Number):
              if isinstance(expt, Float):
                  return self._eval_evalf(expt._prec)**expt
              if expt.is_negative:
                  # (3/4)**-2 -> (4/3)**2
                  ne = -expt
                  if (ne is S.One):
                      return Rational(self.q, self.p)
                  if self < 0:
                      if expt.q != 1:
                          return -(S.NegativeOne)**((expt.p % expt.q) / \
                                 S(expt.q))*Rational(self.q, -self.p)**ne
                      else:
                          return S.NegativeOne**ne*Rational(self.q, -self.p)**ne
                  else:
                      return Rational(self.q, self.p)**ne
              if expt is S.Infinity: # -oo already caught by test for negative
                  if self.p > self.q:
                      # (3/2)**oo -> oo
                      return S.Infinity
                  if self.p < -self.q:
                      # (-3/2)**oo -> oo + I*oo
                      return S.Infinity + S.Infinity*S.ImaginaryUnit
                  return S.Zero
              if isinstance(expt, Integer):
                  # (4/3)**2 -> 4**2 / 3**2
                  return Rational(self.p**expt.p, self.q**expt.p)
              if isinstance(expt, Rational):
                  if self.p != 1:
                      # (4/3)**(5/6) -> 4**(5/6)*3**(-5/6)
                      return Integer(self.p)**expt*Integer(self.q)**(-expt)
                  # as the above caught negative self.p, now self is positive
                  return Integer(self.q)**Rational(expt.p*(expt.q-1), expt.q) / \
                         Integer(self.q)**Integer(expt.p)
  
          if self.is_negative and expt.is_even:
              return (-self)**expt
  
          return
  
      def _as_mpf_val(self, prec):
          return mlib.from_rational(self.p, self.q, prec, rnd)
  
      def _mpmath_(self, prec, rnd):
          return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd))
  
      def __abs__(self):
          return Rational(abs(self.p), self.q)
  
      def __int__(self):
          p, q = self.p, self.q
          if p < 0:
              return -(-p//q)
          return p//q
  
      def __eq__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy != other  -->  not ==
          if isinstance(other, NumberSymbol):
              if other.is_irrational:
                  return False
              return other.__eq__(self)
          if isinstance(other, Number):
              if isinstance(other, Float):
                  return mlib.mpf_eq(self._as_mpf_val(other._prec), other._mpf_)
              elif isinstance(other, Rational):
                  return self.p == other.p and self.q == other.q
          return False
  
      def __ne__(self, other):
          return not self.__eq__(other)
  
      def __gt__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  --> not <
          if isinstance(other, NumberSymbol):
              return other.__le__(self)
          if other.is_real and other.is_number and not isinstance(other, Rational):
              other = other.evalf()
          if isinstance(other, Number):
              if isinstance(other, Float):
                  return bool(mlib.mpf_gt(
                      self._as_mpf_val(other._prec), other._mpf_))
              return bool(self.p*other.q > self.q*other.p)
          return Expr.__gt__(self, other)
  
      def __ge__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  -->  not <=
          if isinstance(other, NumberSymbol):
              return other.__lt__(self)
          if other.is_real and other.is_number and not isinstance(other, Rational):
              other = other.evalf()
          if isinstance(other, Number):
              if isinstance(other, Float):
                  return bool(mlib.mpf_ge(
                      self._as_mpf_val(other._prec), other._mpf_))
              return bool(self.p*other.q >= self.q*other.p)
          return Expr.__ge__(self, other)
  
  
      def __lt__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  --> not <
          if isinstance(other, NumberSymbol):
              return other.__ge__(self)
          if other.is_real and other.is_number and not isinstance(other, Rational):
              other = other.evalf()
          if isinstance(other, Number):
              if isinstance(other, Float):
                  return bool(mlib.mpf_lt(
                      self._as_mpf_val(other._prec), other._mpf_))
              return bool(self.p*other.q < self.q*other.p)
          return Expr.__lt__(self, other)
  
      def __le__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  -->  not <=
          if isinstance(other, NumberSymbol):
              return other.__gt__(self)
          if other.is_real and other.is_number and not isinstance(other, Rational):
              other = other.evalf()
          if isinstance(other, Number):
              if isinstance(other, Float):
                  return bool(mlib.mpf_le(
                      self._as_mpf_val(other._prec), other._mpf_))
              return bool(self.p*other.q <= self.q*other.p)
          return Expr.__le__(self, other)
  
      def __hash__(self):
          return super(Rational, self).__hash__()
  
      def factors(self, limit=None, use_trial=True,
                                    use_rho=False,
                                    use_pm1=False,
                                    verbose=False,
                                    visual=False):
          """A wrapper to factorint which return factors of self that are
          smaller than limit (or cheap to compute). Special methods of
          factoring are disabled by default so that only trial division is used.
          """
          from sympy.ntheory import factorint
  
          f = factorint(self.p, limit=limit,
                                use_trial=use_trial,
                                use_rho=use_rho,
                                use_pm1=use_pm1,
                                verbose=verbose).copy()
          f = defaultdict(int, f)
          for p, e in factorint(self.q, limit=limit,
                                use_trial=use_trial,
                                use_rho=use_rho,
                                use_pm1=use_pm1,
                                verbose=verbose).items():
              f[p] += -e
  
          if len(f) > 1 and 1 in f:
              del f[1]
          if not f:
              f = {1: 1}
          if not visual:
              return dict(f)
          else:
              if -1 in f:
                  f.pop(-1)
                  args = [S.NegativeOne]
              else:
                  args = []
              args.extend([Pow(*i, **{'evaluate':False})
                           for i in sorted(f.items())])
              return Mul(*args, **{'evaluate':False})
  
      def gcd(self, other):
          """Compute greatest common divisor of input arguments. """
          if type(other) in (int, long):
              p = igcd(self.p, other)
  
              if self.is_Integer:
                  return Integer(p)
              else:
                  return Rational(p, self.q)
          else:
              other = _sympify(other)
  
              if other.is_Rational:
                  p = igcd(self.p, other.p)
  
                  if other.is_Integer:
                      if self.is_Integer:
                          return Integer(p)
                      else:
                          return Rational(p, self.q)
                  else:
                      if self.is_Integer:
                          return Rational(p, other.q)
                      else:
                          return Rational(p, ilcm(self.q, other.q))
              elif other.is_Number:
                  return S.One
              else:
                  raise TypeError("expected integer or rational, got %s" % other)
  
      def lcm(self, other):
          """Compute least common multiple of input arguments. """
          if type(other) in (int, long):
              return Integer(ilcm(self.p, other))
          else:
              other = _sympify(other)
  
              if other.is_Rational:
                  p = ilcm(self.p, other.p)
  
                  if self.is_Integer or other.is_Integer:
                      return Integer(p)
                  else:
                      return Rational(p, igcd(self.q, other.q))
              elif other.is_Number:
                  return self*other
              else:
                  raise TypeError("expected integer or rational, got %s" % other)
  
      def cofactors(self, other):
          """Compute GCD and cofactors of input arguments. """
          other = _sympify(other)
          gcd = self.gcd(other)
  
          if gcd is S.One:
              return gcd, self, other
          else:
              return gcd, self/gcd, other/gcd
  
      def as_numer_denom(self):
          return Integer(self.p), Integer(self.q)
  
      def _sage_(self):
          import sage.all as sage
          return sage.Integer(self.p)/sage.Integer(self.q)
  
      def as_content_primitive(self, radical=False):
          """Return the tuple (R, self/R) where R is the positive Rational
          extracted from self.
  
          Examples
          ========
  
          >>> from sympy import S
          >>> (S(-3)/2).as_content_primitive()
          (3/2, -1)
  
          See docstring of Expr.as_content_primitive for more examples.
          """
  
          if self:
              if self.is_positive:
                  return self, S.One
              return -self, S.NegativeOne
          return S.One, self
  
  # int -> Integer
  _intcache = {}
  
  
  # TODO move this tracing facility to sympy/core/trace.py  ?
  def _intcache_printinfo():
      ints = sorted(_intcache.keys())
      nhit = _intcache_hits
      nmiss= _intcache_misses
  
      if nhit == 0 and nmiss == 0:
          print
          print 'Integer cache statistic was not collected'
          return
  
      miss_ratio = float(nmiss) / (nhit+nmiss)
  
      print
      print 'Integer cache statistic'
      print '-----------------------'
      print
      print '#items: %i' % len(ints)
      print
      print ' #hit   #miss               #total'
      print
      print '%5i   %5i (%7.5f %%)   %5i' % (nhit, nmiss, miss_ratio*100, nhit + nmiss)
      print
      print ints
  
  _intcache_hits = 0
  _intcache_misses = 0
  
  def int_trace(f):
      import os
      if os.getenv('SYMPY_TRACE_INT', 'no').lower() != 'yes':
          return f
  
      def Integer_tracer(cls, i):
          global _intcache_hits, _intcache_misses
  
          try:
              _intcache_hits += 1
              return _intcache[i]
          except KeyError:
              _intcache_hits -= 1
              _intcache_misses += 1
  
              return f(cls, i)
  
  
      # also we want to hook our _intcache_printinfo into sys.atexit
      import atexit
      atexit.register(_intcache_printinfo)
  
      return Integer_tracer
  
  
  
  
  class Integer(Rational):
  
      q = 1
      is_integer = True
  
      is_Integer = True
  
      __slots__ = ['p']
  
      def _as_mpf_val(self, prec):
          return mlib.from_int(self.p, prec)
  
      def _mpmath_(self, prec, rnd):
          return mpmath.make_mpf(self._as_mpf_val(prec))
  
      # TODO caching with decorator, but not to degrade performance
      @int_trace
      def __new__(cls, i):
          if isinstance(i, basestring):
              i = i.replace(' ', '')
          # whereas we cannot, in general, make a Rational from an
          # arbitrary expression, we can make an Integer unambiguously
          # (except when a non-integer expression happens to round to
          # an integer). So we proceed by taking int() of the input and
          # let the int routines determine whether the expression can
          # be made into an int or whether an error should be raised.
          ival = int(i)
          try:
              return _intcache[ival]
          except KeyError:
              # We only work with well-behaved integer types. This converts, for
              # example, numpy.int32 instances.
              obj = Expr.__new__(cls)
              obj.p = ival
  
              _intcache[ival] = obj
              return obj
  
      def __getnewargs__(self):
          return (self.p,)
  
      # Arithmetic operations are here for efficiency
      def __int__(self):
          return self.p
  
      def __neg__(self):
          return Integer(-self.p)
  
      def __abs__(self):
          if self.p >= 0:
              return self
          else:
              return Integer(-self.p)
  
      def __divmod__(self, other):
          from containers import Tuple
          if isinstance(other, Integer):
              return Tuple(*(divmod(self.p, other.p)))
          else:
              return Number.__divmod__(self, other)
  
      def __rdivmod__(self, other):
          from containers import Tuple
          if isinstance(other, (int, long)):
              return Tuple(*(divmod(other, self.p)))
          else:
              try:
                  other = Number(other)
              except TypeError:
                  msg = "unsupported operand type(s) for divmod(): '%s' and '%s'"
                  oname = type(other).__name__
                  sname = type(self).__name__
                  raise TypeError(msg % (oname, sname))
              return Number.__divmod__(other, self)
  
      # TODO make it decorator + bytecodehacks?
      def __add__(self, other):
          if isinstance(other, (int, long)):
              return Integer(self.p + other)
          elif isinstance(other, Integer):
              return Integer(self.p + other.p)
          return Rational.__add__(self, other)
  
      def __radd__(self, other):
          if isinstance(other, (int, long)):
              return Integer(other + self.p)
          return Rational.__add__(self, other)
  
      def __sub__(self, other):
          if isinstance(other, (int, long)):
              return Integer(self.p - other)
          elif isinstance(other, Integer):
              return Integer(self.p - other.p)
          return Rational.__sub__(self, other)
  
      def __rsub__(self, other):
          if isinstance(other, (int, long)):
              return Integer(other - self.p)
          return Rational.__rsub__(self, other)
  
      def __mul__(self, other):
          if isinstance(other, (int, long)):
              return Integer(self.p*other)
          elif isinstance(other, Integer):
              return Integer(self.p*other.p)
          return Rational.__mul__(self, other)
  
      def __rmul__(self, other):
          if isinstance(other, (int, long)):
              return Integer(other*self.p)
          return Rational.__mul__(self, other)
  
      def __mod__(self, other):
          if isinstance(other, (int, long)):
              return Integer(self.p % other)
          elif isinstance(other, Integer):
              return Integer(self.p % other.p)
          return Rational.__mod__(self, other)
  
      def __rmod__(self, other):
          if isinstance(other, (int, long)):
              return Integer(other % self.p)
          elif isinstance(other, Integer):
              return Integer(other.p % self.p)
          return Rational.__rmod__(self, other)
  
      def __eq__(self, other):
          if isinstance(other, (int, long)):
              return (self.p == other)
          elif isinstance(other, Integer):
              return (self.p == other.p)
          return Rational.__eq__(self, other)
  
      def __ne__(self, other):
          return not self.__eq__(other)
  
      def __gt__(self, other):
          if isinstance(other, (int, long)):
              return (self.p >  other)
          elif isinstance(other, Integer):
              return (self.p >  other.p)
          return Rational.__gt__(self, other)
  
      def __lt__(self, other):
          if isinstance(other, (int, long)):
              return (self.p <  other)
          elif isinstance(other, Integer):
              return (self.p <  other.p)
          return Rational.__lt__(self, other)
  
      def __ge__(self, other):
          if isinstance(other, (int, long)):
              return (self.p >= other)
          elif isinstance(other, Integer):
              return (self.p >= other.p)
          return Rational.__ge__(self, other)
  
      def __le__(self, other):
          if isinstance(other, (int, long)):
              return (self.p <= other)
          elif isinstance(other, Integer):
              return (self.p <= other.p)
          return Rational.__le__(self, other)
  
      def __hash__(self):
          return super(Integer, self).__hash__()
  
      def __index__(self):
          return self.p
  
      ########################################
  
      def _eval_is_odd(self):
          return bool(self.p % 2)
  
      def _eval_power(self, expt):
          """
          Tries to do some simplifications on self**expt
  
          Returns None if no further simplifications can be done
  
          When exponent is a fraction (so we have for example a square root),
          we try to find a simpler representation by factoring the argument
          up to factors of 2**15, e.g.
  
            - sqrt(4) becomes 2
            - sqrt(-4) becomes 2*I
            - (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7)
  
          Further simplification would require a special call to factorint on
          the argument which is not done here for sake of speed.
  
          """
          from sympy import perfect_power
  
          if expt is S.Infinity:
              if self.p > S.One:
                  return S.Infinity
              # cases -1, 0, 1 are done in their respective classes
              return S.Infinity + S.ImaginaryUnit*S.Infinity
          if expt is S.NegativeInfinity:
              return Rational(1, self)**S.Infinity
          if not isinstance(expt, Number):
              # simplify when expt is even
              # (-2)**k --> 2**k
              if self.is_negative and expt.is_even:
                  return (-self)**expt
          if not isinstance(expt, Rational):
              return
          if expt is S.Half and self < 0:
              # we extract I for this special case since everyone is doing so
              return S.ImaginaryUnit*Pow(-self, expt)
          if expt < 0:
              # invert base and change sign on exponent
              ne = -expt
              if self < 0:
                  if expt.q != 1:
                      return -(S.NegativeOne)**((expt.p % expt.q) / \
                              S(expt.q))*Rational(1, -self)**ne
                  else:
                      return (S.NegativeOne)**ne*Rational(1, -self)**ne
              else:
                  return Rational(1, self.p)**ne
          # see if base is a perfect root, sqrt(4) --> 2
          x, xexact = integer_nthroot(abs(self.p), expt.q)
          if xexact:
              # if it's a perfect root we've finished
              result = Integer(x**abs(expt.p))
              if self < 0:
                  result*= (-1)**expt
              return result
  
          # The following is an algorithm where we collect perfect roots
          # from the factors of base.
  
          # if it's not an nth root, it still might be a perfect power
          b_pos = int(abs(self.p))
          p = perfect_power(b_pos)
          if p is not False:
              dict = {p[0]: p[1]}
          else:
              dict = Integer(self).factors(limit=2**15)
  
          # now process the dict of factors
          if self.is_negative:
              dict[-1] = 1
          out_int = 1 # integer part
          out_rad = 1 # extracted radicals
          sqr_int = 1
          sqr_gcd = 0
          sqr_dict = {}
          for prime, exponent in dict.items():
              exponent *= expt.p
              # remove multiples of expt.q: (2**12)**(1/10) -> 2*(2**2)**(1/10)
              div_e, div_m = divmod(exponent, expt.q)
              if div_e > 0:
                  out_int *= prime**div_e
              if div_m > 0:
                  # see if the reduced exponent shares a gcd with e.q
                  # (2**2)**(1/10) -> 2**(1/5)
                  g = igcd(div_m, expt.q)
                  if g != 1:
                      out_rad *= Pow(prime, Rational(div_m//g, expt.q//g))
                  else:
                      sqr_dict[prime] = div_m
          # identify gcd of remaining powers
          for p, ex in sqr_dict.iteritems():
              if sqr_gcd == 0:
                  sqr_gcd = ex
              else:
                  sqr_gcd = igcd(sqr_gcd, ex)
                  if sqr_gcd == 1:
                      break
          for k, v in sqr_dict.iteritems():
              sqr_int *= k**(v//sqr_gcd)
          if sqr_int == self and out_int == 1 and out_rad == 1:
              result = None
          else:
              result = out_int*out_rad*Pow(sqr_int, Rational(sqr_gcd, expt.q))
          return result
  
      def _eval_is_prime(self):
          from sympy.ntheory import isprime
  
          return isprime(self)
  
      def as_numer_denom(self):
          return self, S.One
  
      def __floordiv__(self, other):
          return Integer(self.p // Integer(other).p)
  
      def __rfloordiv__(self, other):
          return Integer(Integer(other).p // self.p)
  
      def factorial(self):
          """Compute factorial of `self`. """
          from sympy.functions.combinatorial.factorials import factorial
          return Integer(factorial(int(self)))
  
      def isqrt(self):
          """Compute integer square root of `self`. """
          return Integer(mlib.isqrt(int(self)))
  
      def half_gcdex(self, other):
          """Half Extended Euclidean Algorithm. """
          s, _, h = self.gcdex(other)
          return s, h
  
      def gcdex(self, other):
          """Extended Euclidean Algorithm. """
          if isinstance(other, (int, long)):
              return tuple(map(Integer, igcdex(int(self), other)))
          b = _sympify(other)
          if b.is_Integer:
              return tuple(map(Integer, igcdex(int(self), int(b))))
          else:
              raise ValueError("expected an integer, got %s" % b)
  
      def invert(self, other):
          """Invert `self` modulo `other`, if possible. """
          if isinstance(other, (int, long)):
              a, b = int(self), other
          else:
              b = _sympify(other)
  
              if b.is_Integer:
                  a, b = int(self), int(b)
              else:
                  raise ValueError("expected an integer, got %s" % b)
  
          s, _, h = igcdex(a, b)
  
          if h == 1:
              return Integer(s % b)
          else:
              raise ZeroDivisionError("zero divisor")
  
  # Add sympify converters
  converter[int] = converter[long] = Integer
  
  class RationalConstant(Rational):
      """
      Abstract base class for rationals with specific behaviors
  
      Derived classes must define class attributes p and q and should probably all
      be singletons.
      """
      __slots__ = []
  
      def __new__(cls):
          return AtomicExpr.__new__(cls)
  
  class IntegerConstant(Integer):
      __slots__ = []
  
      def __new__(cls):
          return AtomicExpr.__new__(cls)
  
  
  class Zero(IntegerConstant):
      __metaclass__ = Singleton
  
      p = 0
      q = 1
      is_positive = False
      is_negative = False
      is_finite = False
      is_zero = True
      is_composite = False
  
      __slots__ = []
  
      @staticmethod
      def __abs__():
          return S.Zero
  
      @staticmethod
      def __neg__():
          return S.Zero
  
      @_sympifyit('other', NotImplemented)
      def __mul__(self, other):
          if other is S.NaN or \
             other is S.NegativeInfinity or \
             other is S.Infinity or \
             other is S.ComplexInfinity:
              return S.NaN
          return S.Zero
  
      def _eval_power(self, expt):
          if expt.is_positive:
              return self
          if expt.is_negative:
              return S.Infinity
          if expt.is_real is False:
              return S.NaN
          # infinities are already handled with pos and neg
          # tests above; now throw away leading numbers on Mul
          # exponent
          coeff, terms = expt.as_coeff_Mul()
          if coeff.is_negative:
              return S.Infinity**terms
          if coeff is not S.One: # there is a Number to discard
              return self**terms
  
      def _eval_order(self, *symbols):
          # Order(0,x) -> 0
          return self
  
      def __nonzero__(self):
          return False
  
  class One(IntegerConstant):
      __metaclass__ = Singleton
  
      p = 1
      q = 1
  
      __slots__ = []
  
      @staticmethod
      def __abs__():
          return S.One
  
      @staticmethod
      def __neg__():
          return S.NegativeOne
  
      def _eval_power(self, expt):
          return self
  
      def _eval_order(self, *symbols):
          return
  
      @staticmethod
      def factors(limit=None, use_trial=True,
                              use_rho=False,
                              use_pm1=False,
                              verbose=False,
                              visual=False):
          if visual:
              return S.One
          return {1: 1}
  
  class NegativeOne(IntegerConstant):
      __metaclass__ = Singleton
  
      p = -1
      q = 1
  
      __slots__ = []
  
      @staticmethod
      def __abs__():
          return S.One
  
      @staticmethod
      def __neg__():
          return S.One
  
      def _eval_power(self, expt):
          if expt.is_odd:
              return S.NegativeOne
          if expt.is_even:
              return S.One
          if isinstance(expt, Number):
              if isinstance(expt, Float):
                  return Float(-1.0)**expt
              if expt is S.NaN:
                  return S.NaN
              if expt is S.Infinity or expt is S.NegativeInfinity:
                  return S.NaN
              if expt is S.Half:
                  return S.ImaginaryUnit
              if isinstance(expt, Rational):
                  if expt.q == 2:
                      return S.ImaginaryUnit**Integer(expt.p)
                  i, r = divmod(expt.p, expt.q)
                  if i:
                      return self**i*self**Rational(r, expt.q)
          return
  
  class Half(RationalConstant):
      __metaclass__ = Singleton
  
      p = 1
      q = 2
  
      __slots__ = []
  
      @staticmethod
      def __abs__():
          return S.Half
  
  
  class Infinity(Number):
      __metaclass__ = Singleton
  
      is_commutative   = True
      is_positive      = True
      is_bounded       = False
      is_finite        = False
      is_infinitesimal = False
      is_integer       = None
      is_rational      = None
      is_odd           = None
  
      __slots__ = []
  
      def __new__(cls):
          return AtomicExpr.__new__(cls)
  
      @_sympifyit('other', NotImplemented)
      def __add__(self, other):
          if isinstance(other, Number):
              if other is S.NegativeInfinity or other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other == Float('-inf') or other._mpf_ == fnan:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  else:
                      return Float('inf')
              else:
                  return S.Infinity
          return NotImplemented
      __radd__ = __add__
  
      @_sympifyit('other', NotImplemented)
      def __sub__(self, other):
          if isinstance(other, Number):
              if other is S.Infinity or other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other == Float('inf') or other._mpf_ == fnan:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  else:
                      return Float('inf')
              else:
                  return S.Infinity
          return NotImplemented
  
      @_sympifyit('other', NotImplemented)
      def __mul__(self, other):
          if isinstance(other, Number):
              if other is S.Zero or other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other._mpf_ == fnan or other == 0:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  if other > 0:
                      return Float('inf')
                  else:
                      return Float('-inf')
              else:
                  if other > 0:
                      return S.Infinity
                  else:
                      return S.NegativeInfinity
          return NotImplemented
      __rmul__ = __mul__
  
      @_sympifyit('other', NotImplemented)
      def __div__(self, other):
          if isinstance(other, Number):
              if other is S.Infinity or \
                 other is S.NegativeInfinity or \
                 other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other == Float('-inf') or \
                     other == Float('inf') or \
                     other._mpf_ == fnan:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  elif other >= 0:
                      return Float('inf')
                  else:
                      return Float('-inf')
              else:
                  if other >= 0:
                      return S.Infinity
                  else:
                      return S.NegativeInfinity
          return NotImplemented
  
      __truediv__ = __div__
  
      def __abs__(self):
          return S.Infinity
  
      def __neg__(self):
          return S.NegativeInfinity
  
      def _eval_power(self, expt):
          """
          ``expt`` is symbolic object but not equal to 0 or 1.
  
          ================ ======= ==============================
          Expression       Result  Notes
          ================ ======= ==============================
          ``oo ** nan``    ``nan``
          ``oo ** -p``     ``0``   ``p`` is number, ``oo``
          ================ ======= ==============================
  
          """
          if expt.is_positive:
              return S.Infinity
          if expt.is_negative:
              return S.Zero
          if expt is S.NaN:
              return S.NaN
  
          if expt.is_number:
              return self**expt.evalf()
  
      def _as_mpf_val(self, prec):
          return mlib.finf
  
      def _sage_(self):
          import sage.all as sage
          return sage.oo
  
      def __hash__(self):
          return super(Infinity, self).__hash__()
  
      def __eq__(self, other):
          return other is S.Infinity
  
      def __ne__(self, other):
          return other is not S.Infinity
  
      def __lt__(self, other):
          return False
  
      def __le__(self, other):
          return other is S.Infinity
  
      def __gt__(self, other):
          return other is not S.Infinity
  
      def __ge__(self, other):
          return True
  
      def __mod__(self, other):
          return S.NaN
  
      __rmod__ = __mod__
  
  oo = S.Infinity
  
  class NegativeInfinity(Number):
      __metaclass__ = Singleton
  
      is_commutative   = True
      is_real          = True
      is_positive      = False
      is_bounded       = False
      is_finite        = False
      is_infinitesimal = False
      is_integer       = None
      is_rational      = None
  
      __slots__ = []
  
      def __new__(cls):
          return AtomicExpr.__new__(cls)
  
      @_sympifyit('other', NotImplemented)
      def __add__(self, other):
          if isinstance(other, Number):
              if other is S.Infinity or other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other == Float('inf') or other._mpf_ == fnan:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  else:
                      return Float('-inf')
              else:
                  return S.NegativeInfinity
          return NotImplemented
      __radd__ = __add__
  
      @_sympifyit('other', NotImplemented)
      def __sub__(self, other):
          if isinstance(other, Number):
              if other is S.NegativeInfinity or other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other == Float('-inf') or other._mpf_ == fnan:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  else:
                      return Float('-inf')
              else:
                  return S.NegativeInfinity
          return NotImplemented
  
      @_sympifyit('other', NotImplemented)
      def __mul__(self, other):
          if isinstance(other, Number):
              if other is S.Zero or other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other._mpf_ == fnan or other == 0:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  elif other > 0:
                      return Float('-inf')
                  else:
                      return Float('inf')
              else:
                  if other > 0:
                      return S.NegativeInfinity
                  else:
                      return S.Infinity
          return NotImplemented
      __rmul__ = __mul__
  
      @_sympifyit('other', NotImplemented)
      def __div__(self, other):
          if isinstance(other, Number):
              if other is S.Infinity or \
                 other is S.NegativeInfinity or \
                 other is S.NaN:
                  return S.NaN
              elif other.is_Float:
                  if other == Float('-inf') or \
                     other == Float('inf') or \
                     other == Float('nan') or \
                     other._mpf_ == fnan:
                      #Used workaround because Float('nan') == Float('nan') return False
                      return Float('nan')
                  elif other >= 0:
                      return Float('-inf')
                  else:
                      return Float('inf')
              else:
                  if other >= 0:
                      return S.NegativeInfinity
                  else:
                      return S.Infinity
          return NotImplemented
  
      __truediv__ = __div__
  
      def __abs__(self):
          return S.Infinity
  
      def __neg__(self):
          return S.Infinity
  
      def _eval_power(self, expt):
          """
          ``expt`` is symbolic object but not equal to 0 or 1.
  
          ================ ======= ==============================
          Expression       Result  Notes
          ================ ======= ==============================
          ``(-oo) ** nan`` ``nan``
          ``(-oo) ** oo``  ``nan``
          ``(-oo) ** -oo`` ``nan``
          ``(-oo) ** e``   ``oo``  ``e`` is positive even integer
          ``(-oo) ** o``   ``-oo`` ``o`` is positive odd integer
          ================ ======= ==============================
  
          """
          if isinstance(expt, Number):
              if expt is S.NaN or \
                 expt is S.Infinity or \
                 expt is S.NegativeInfinity:
                  return S.NaN
  
              if isinstance(expt, Integer) and expt.is_positive:
                  if expt.is_odd:
                      return S.NegativeInfinity
                  else:
                      return S.Infinity
  
              return S.NegativeOne**expt*S.Infinity**expt
  
      def _as_mpf_val(self, prec):
          return mlib.fninf
  
      def _sage_(self):
          import sage.all as sage
          return -(sage.oo)
  
      def __hash__(self):
          return super(NegativeInfinity, self).__hash__()
  
      def __eq__(self, other):
          return other is S.NegativeInfinity
  
      def __ne__(self, other):
          return other is not S.NegativeInfinity
  
      def __lt__(self, other):
          return other is not S.NegativeInfinity
  
      def __le__(self, other):
          return True
  
      def __gt__(self, other):
          return False
  
      def __ge__(self, other):
          return other is S.NegativeInfinity
  
  class NaN(Number):
      """
      Not a Number.
  
      This represents the corresponding data type to floating point nan, which
      is defined in the IEEE 754 floating point standard, and corresponds to the
      Python ``float('nan')``.
  
      NaN serves as a place holder for numeric values that are indeterminate,
      but not infinite.  Most operations on nan, produce another nan.  Most
      indeterminate forms, such as ``0/0`` or ``oo - oo` produce nan.  Three
      exceptions are ``0**0``, ``1**oo``, and ``oo**0``, which all produce ``1``
      (this is consistent with Python's float).
  
      NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported
      as ``nan``.
  
      Examples
      ========
  
      >>> from sympy import nan, S, oo
      >>> nan is S.NaN
      True
      >>> oo - oo
      nan
      >>> nan + 1
      nan
  
      References
      ==========
  
      - http://en.wikipedia.org/wiki/NaN
  
      """
      __metaclass__ = Singleton
  
      is_commutative = True
      is_real = None
      is_rational = None
      is_integer = None
      is_comparable = False
      is_finite = None
      is_bounded = None
      is_zero = None
      is_prime = None
      is_positive = None
      is_negative = None
  
      __slots__ = []
  
      def __new__(cls):
          return AtomicExpr.__new__(cls)
  
      @_sympifyit('other', NotImplemented)
      def __add__(self, other):
          return self
  
      @_sympifyit('other', NotImplemented)
      def __sub__(self, other):
          return self
  
      @_sympifyit('other', NotImplemented)
      def __mul__(self, other):
          return self
  
      @_sympifyit('other', NotImplemented)
      def __div__(self, other):
          return self
  
      __truediv__ = __div__
  
      def _as_mpf_val(self, prec):
          return mlib.fnan
  
      def _sage_(self):
          import sage.all as sage
          return sage.NaN
  
      def __hash__(self):
          return super(NaN, self).__hash__()
  
      def __eq__(self, other):
          return other is S.NaN
  
      def __ne__(self, other):
          return other is not S.NaN
  
      def __gt__(self, other):
          return False
  
      def __ge__(self, other):
          return False
  
      def __lt__(self, other):
          return False
  
      def __le__(self, other):
          return False
  
  nan = S.NaN
  
  class ComplexInfinity(AtomicExpr):
      __metaclass__ = Singleton
  
      is_commutative = True
      is_bounded = False
      is_real = None
      is_number = False
  
      __slots__ = []
  
      def __new__(cls):
          return AtomicExpr.__new__(cls)
  
      @staticmethod
      def __abs__():
          return S.Infinity
  
      @staticmethod
      def __neg__():
          return S.ComplexInfinity
  
      def _eval_power(self, expt):
          if expt is S.ComplexInfinity:
              return S.NaN
  
          if isinstance(expt, Number):
              if expt is S.Zero:
                  return S.NaN
              else:
                  if expt.is_positive:
                      return S.ComplexInfinity
                  else:
                      return S.Zero
  
  zoo = S.ComplexInfinity
  
  class NumberSymbol(AtomicExpr):
      __metaclass__ = Singleton
  
      is_commutative = True
      is_bounded = True
      is_finite = True
      is_number = True
  
      __slots__ = []
  
      is_NumberSymbol = True
  
      def __new__(cls):
          return AtomicExpr.__new__(cls)
  
      def approximation(self, number_cls):
          """ Return an interval with number_cls endpoints
          that contains the value of NumberSymbol.
          If not implemented, then return None.
          """
  
      def _eval_evalf(self, prec):
          return Float._new(self._as_mpf_val(prec), prec)
  
      def __eq__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy != other  -->  not ==
          if self is other:
              return True
          if isinstance(other, Number) and self.is_irrational:
              return False
  
          return False    # NumberSymbol != non-(Number|self)
  
      def __ne__(self, other):
          return not self.__eq__(other)
  
      def __lt__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  --> not <
          if self is other:
              return False
          if isinstance(other, Number):
              approx = self.approximation_interval(other.__class__)
              if approx is not None:
                  l,u = approx
                  if other < l:
                      return False
                  if other > u:
                      return True
              return self.evalf() < other
          if other.is_real and other.is_number:
              other = other.evalf()
              return self.evalf() < other
          return Expr.__lt__(self, other)
  
      def __le__(self, other):
          try:
              other = _sympify(other)
          except SympifyError:
              return False    # sympy > other  --> not <=
          if self is other:
              return True
          if other.is_real and other.is_number:
              other = other.evalf()
          if isinstance(other, Number):
              return self.evalf() <= other
          return Expr.__le__(self, other)
  
      def __gt__(self, other):
          return (-self) < (-other)
  
      def __ge__(self, other):
          return (-self) <= (-other)
  
      def __int__(self):
          # subclass with appropriate return value
          raise NotImplementedError
  
      def __hash__(self):
          return super(NumberSymbol, self).__hash__()
  
  
  class Exp1(NumberSymbol):
      __metaclass__ = Singleton
  
      is_real = True
      is_positive = True
      is_negative = False # XXX Forces is_negative/is_nonnegative
      is_irrational = True
  
      __slots__ = []
  
      @staticmethod
      def __abs__():
          return S.Exp1
  
      def __int__(self):
          return 2
  
      def _as_mpf_val(self, prec):
          return mpf_e(prec)
  
      def approximation_interval(self, number_cls):
          if issubclass(number_cls,Integer):
              return (Integer(2),Integer(3))
          elif issubclass(number_cls,Rational):
              pass
  
      def _eval_power(self, expt):
          return C.exp(expt)
  
      def _sage_(self):
          import sage.all as sage
          return sage.e
  E = S.Exp1
  
  class Pi(NumberSymbol):
      __metaclass__ = Singleton
  
  
      is_real = True
      is_positive = True
      is_negative = False
      is_irrational = True
  
      __slots__ = []
  
      @staticmethod
      def __abs__():
          return S.Pi
  
      def __int__(self):
          return 3
  
      def _as_mpf_val(self, prec):
          return mpf_pi(prec)
  
      def approximation_interval(self, number_cls):
          if issubclass(number_cls, Integer):
              return (Integer(3), Integer(4))
          elif issubclass(number_cls, Rational):
              return (Rational(223,71), Rational(22,7))
  
      def _sage_(self):
          import sage.all as sage
          return sage.pi
  pi = S.Pi
  
  class GoldenRatio(NumberSymbol):
      __metaclass__ = Singleton
  
      is_real = True
      is_positive = True
      is_negative = False
      is_irrational = True
  
      __slots__ = []
  
      def __int__(self):
          return 1
  
      def _as_mpf_val(self, prec):
           # XXX track down why this has to be increased
          rv = mlib.from_man_exp(phi_fixed(prec+10), -prec-10)
          return mpf_norm(rv, prec)
  
      def _eval_expand_func(self, **hints):
          from sympy import sqrt
          return S.Half + S.Half*sqrt(5)
  
      def approximation_interval(self, number_cls):
          if issubclass(number_cls, Integer):
              return (S.One, Rational(2))
          elif issubclass(number_cls, Rational):
              pass
  
      def _sage_(self):
          import sage.all as sage
          return sage.golden_ratio
  
  class EulerGamma(NumberSymbol):
      __metaclass__ = Singleton
  
      is_real = True
      is_positive = True
      is_negative = False
      is_irrational = None
  
      __slots__ = []
  
      def __int__(self):
          return 0
  
      def _as_mpf_val(self, prec):
           # XXX track down why this has to be increased
          v = mlib.libhyper.euler_fixed(prec+10)
          rv = mlib.from_man_exp(v, -prec-10)
          return mpf_norm(rv, prec)
  
      def approximation_interval(self, number_cls):
          if issubclass(number_cls, Integer):
              return (S.Zero, S.One)
          elif issubclass(number_cls, Rational):
              return (S.Half, Rational(3, 5))
  
      def _sage_(self):
          import sage.all as sage
          return sage.euler_gamma
  
  class Catalan(NumberSymbol):
      __metaclass__ = Singleton
  
      is_real = True
      is_positive = True
      is_negative = False
      is_irrational = None
  
      __slots__ = []
  
      def __int__(self):
          return 0
  
      def _as_mpf_val(self, prec):
          # XXX track down why this has to be increased
          v = mlib.catalan_fixed(prec+10)
          rv = mlib.from_man_exp(v, -prec-10)
          return mpf_norm(rv, prec)
  
      def approximation_interval(self, number_cls):
          if issubclass(number_cls, Integer):
              return (S.Zero, S.One)
          elif issubclass(number_cls, Rational):
              return (Rational(9, 10), S.One)
  
      def _sage_(self):
          import sage.all as sage
          return sage.catalan
  
  class ImaginaryUnit(AtomicExpr):
      __metaclass__ = Singleton
  
      is_commutative = True
      is_imaginary = True
      is_bounded = True
      is_finite = True
      is_number = True
  
      __slots__ = []
  
      @staticmethod
      def __abs__():
          return S.One
  
      def _eval_evalf(self, prec):
          return self
  
      def _eval_conjugate(self):
          return -S.ImaginaryUnit
  
      def _eval_power(self, expt):
          """
          b is I = sqrt(-1)
          e is symbolic object but not equal to 0, 1
  
          I**r -> (-1)**(r/2) -> exp(r/2*Pi*I) -> sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal
          I**0 mod 4 -> 1
          I**1 mod 4 -> I
          I**2 mod 4 -> -1
          I**3 mod 4 -> -I
          """
  
  
          if isinstance(expt, Number):
              if isinstance(expt, Integer):
                  expt = expt.p % 4
                  if expt == 0:
                      return S.One
                  if expt == 1:
                      return S.ImaginaryUnit
                  if expt == 2:
                      return -S.One
                  return -S.ImaginaryUnit
              return (S.NegativeOne)**(expt*S.Half)
          return
  
      def as_base_exp(self):
          return S.NegativeOne, S.Half
  
      def _sage_(self):
          import sage.all as sage
          return sage.I
  
  I = S.ImaginaryUnit
  
  try:
      # fractions is only available for python 2.6+
      import fractions
  
      def sympify_fractions(f):
          return Rational(f.numerator, f.denominator)
  
      converter[fractions.Fraction] = sympify_fractions
  except ImportError:
      pass
  
  try:
      import gmpy
  
      def sympify_mpz(x):
          return Integer(long(x))
  
      def sympify_mpq(x):
          return Rational(long(x.numer()), long(x.denom()))
  
      converter[type(gmpy.mpz(1))] = sympify_mpz
      converter[type(gmpy.mpq(1, 2))] = sympify_mpq
  except ImportError:
      pass
  
  def sympify_mpmath(x):
      return Expr._from_mpmath(x, x.context.prec)
  
  converter[mpnumeric] = sympify_mpmath
  
  def sympify_complex(a):
      real, imag = map(sympify, (a.real, a.imag))
      return real + S.ImaginaryUnit*imag
  
  converter[complex] = sympify_complex
  
  _intcache[0] = S.Zero
  _intcache[1] = S.One
  _intcache[-1]= S.NegativeOne
  
  from power import Pow, integer_nthroot
  from mul import Mul
  Mul.identity = One()
  from add import Add
  Add.identity = Zero()
  